18.357 Interfacial Phenomena, Lectures All

18.357 Interfacial Phenomena, Fall 2010

taught by Prof. John W. M. Bush

June 3, 2013

Contents

1 Introduction, Notation 3

1.1 Suggested References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Definition and Scaling of Surface Tension 4

2.1 History: Surface tension in antiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Motivation: Who cares about surface tension? . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Surface tension: a working definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 The scaling of surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.6 A few simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Wetting 11

4 Youngs Law with Applications 12

4.1 Formal Development of Interfacial Flow Problems . . . . . . . . . . . . . . . . . . . . . . . 12

5 Stress Boundary Conditions 14

5.1 Stress conditions at a fluid-fluid interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.2 Appendix A : Useful identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.3 Fluid Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.4 Appendix B : Computing curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.5 Appendix C : Frenet-Serret Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 More on Fluid statics 22

6.1 Capillary forces on floating bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7 Spinning, tumbling and rolling drops 24

7.1 Rotating Drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 7.2 Rolling drops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

8 Capillary Rise 26

8.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9 Marangoni Flows 30

10 Marangoni Flows II 34

10.1 Tears of Wine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 10.2 Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 10.3 Surfactant-induced Marangoni flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 10.4 Bubble motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1

Contents Contents

11 Fluid Jets 40

11.1 The shape of a falling fluid jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 11.2 The Plateau-Rayleigh Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 11.3 Fluid Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12 Instability Dynamics 45

12.1 Capillary Instability of a Fluid Coating on a Fiber . . . . . . . . . . . . . . . . . . . . . . 45 12.2 Dynamics of Instability (Rayleigh 1879) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 12.3 Rupture of a Soap Film (Culick 1960, Taylor 1960) . . . . . . . . . . . . . . . . . . . . . . 47

13 Fluid Sheets 49

13.1 Fluid Sheets: shape and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 13.2 Circular Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 13.3 Lenticular sheets with unstable rims (Taylor 1960) . . . . . . . . . . . . . . . . . . . . . . 51 13.4 Lenticular sheets with stable rims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 13.5 Water Bells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 13.6 Swirling Water Bell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

14 Instability of Superposed Fluids 55

14.1 Rayleigh-Taylor Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 14.2 Kelvin-Helmholtz Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

15 Contact angle hysteresis, Wetting of textured solids 59

15.1 Contact Angle Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 15.2 Wetting of a Rough Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 15.3 Wenzel State (1936) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 15.4 Cassie-Baxter State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

16 More forced wetting 64

16.1 Hydrophobic Case: e > /2, cos e < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 16.2 Hydrophilic Case: e < /2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 16.3 Forced Wetting: the Landau-Levich-Derjaguin Problem . . . . . . . . . . . . . . . . . . . 66

17 Coating: Dynamic Contact Lines 68

17.1 Contact Line Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

18 Spreading 74

18.1 Spreading of small drops on solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 18.2 Immiscible Drops at an Interface Pujado & Scriven 1972 . . . . . . . . . . . . . . . . . . 75 18.3 Oil Spill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 18.4 Oil on water: A brief review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 18.5 The Beating Heart Stocker & Bush (JFM 2007) . . . . . . . . . . . . . . . . . . . . . . . . 77

19 Water waves 79

MIT OCW: 18.357 Interfacial Phenomena 2 Prof. John W. M. Bush

1. Introduction, Notation We consider fluid systems dominated by the influence of interfacial tension. The roles of curvature pressure and Marangoni stress are elucidated in a variety of situtations. Particular attention will be given to the dynamics of drops and bubbles, soap films and minimal surfaces, wetting phenomena, water-repellency, surfactants, Marangoni flows, capillary origami and contact line dynamics. Theoretical developments will be accompanied by classroom demonstrations. The role of surface tension in biology will be highlighted.

Notation

Nomenclature: denotes surface tension (at fluid-gas interface) denotes interfacial tension (at fluid-fluid or fluid-solid interface).

Note on units: we will use predominantly cgs system. 2Unit of force: 1 dyne = 1 g cm s = 105N as the cgs unit of force, roughly the weight of 1 mosquito.

Pressure: 1 atm 100kPa = 105N/m2=106 dynes/cm2 . Units: []=dynes/cm=mN/m.

2What is an interface?: roughness scale , from equality of surface and thermal energy get kT (kT/)1/2 . If scales of experiment, can speak of a smooth interface.

1.1 Suggested References

While this list of relevant textbooks is far from complete, we include it as a source of additional reading for the interested student.

Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves

by P.G. de Gennes, F. Brochard-Wyart and D. Quere. Springer Publishing. A readable and accessible treatment of a wide range of capillary phenomena.

Molecular theory of capillarity

by J.S. Rowlinson and B. Widom. Dover 1982.

Intermolecular and surface forces

by J. Israelachvili. Academic Press, 2nd edition 1995.

Multimedia Fluid Mechanics

Cambridge University Press, Ed. Bud Homsy. A DVD with an extensive section devoted to capillary effects. Relevant videos will be used throughout the course.

3

2. Definition and Scaling of Surface

Tension

These lecture notes have been drawn from many sources, including textbooks, journal articles, and lecture notes from courses taken by the author as a student. These notes are not intended as a complete discussion of the subject, or as a scholarly work in which all relevant references are cited. Rather, they are intended as an introduction that will hopefully motivate the interested student to learn more about the subject. Topics have been chosen according to their perceived value in developing the physical insight of the students.

2.1 History: Surface tension in antiquity

Hero of Alexandria (10 AD - 70 AD) Greek mathematician and engineer, the greatest experimentalist of antiquity. Exploited capillarity in a number of inventions described in his book Pneumatics, including the water clock.

Pliny the Elder (23 AD - 79 AD) Author, natural philosopher, army and naval commander of the early Roman Empire. Described the glassy wakes of ships. True glory comes in doing what deserves to be written; in writing what deserves to be read; and in so living as to make

the world happier. Truth comes out in wine.

Leonardo da Vinci (1452-1519) Reported capillary rise in his notebooks, hypothesized that mountain streams are fed by capillary networks.

Francis Hauksbee (1666-1713) Conducted systematic investigation of capillary rise, his work was described in Newtons Opticks, but no mention was made of him.

Benjamin Franklin (1706-1790) Polymath: scientist, inventor, politician; examined the ability of oil to suppress waves.

Pierre-Simon Laplace (1749-1827) French mathematician and astronomer, elucidated the concept and theoretical description of the meniscus, hence the term Laplace pressure.

Thomas Young (1773-1829) Polymath, solid mechanician, scientist, linguist. Demonstrated the wave nature of light with ripple tank experiments, described wetting of a solid by a fluid.

Joseph Plateau (1801-1883) Belgian physicist, continued his experiments after losing his sight. Extensive study of capillary phenomena, soap films, minimal surfaces, drops and bubbles.

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2.2. Motivation: Who cares about surface tension? Chapter 2. Definition and Scaling of Surface Tension

2.2 Motivation: Who cares about surface tension?

As we shall soon see, surface tension dominates gravity on a scale less than the capillary length (roughly 2mm). It thus plays a critical role in a variety of small-scale processes arising in biology, environmental science and technology.

Biology

all small creatures live in a world dominated by surface tension

surface tension important for insects for many basic functions

weight support and propulsion at the water surface

adhesion and deadhesion via surface tension

the pistol shrimp: hunting with bubbles

underwater breathing and diving via surface tension

natural strategies for water-repellency in plants and animals

the dynamics of lungs and the role of surfactants and impurities

Figure 2.1: The diving bell spider

Geophysics and environmental science

the dynamics of raindrops and their role in the biosphere

most biomaterial is surface active, sticks to the surface of drops / bubbles

chemical, thermal and biological transport in the surf zone

early life: early vessicle formation, confinement to an interface

oil recovery, carbon sequestration, groundwater flows

design of insecticides intended to coat insects, leave plant unharmed

chemical leaching and the water-repellency of soils

oil spill dynamics and mitigation

disease transmission via droplet exhalation

dynamics of magma chambers and volcanoes

the exploding lakes of Cameroon

Technology

capillary effects dominant in microgravity settings: NASA

cavitation-induced damage on propellers and submarines

cavitation in medicine: used to damage kidney stones, tumours ...

design of superhydrophobic surfaces e.g. self-cleaning windows, drag-reducing or erosion-resistant surfaces

lab-on-a-chip technology: medical diagnostics, drug delivery

microfluidics: continuous and discrete fluid transport and mixing

tracking submarines with their surface signature

inkjet printing

the bubble computer


This image has been removed due to copyright restrictions.

Please see the image on page http://news.sciencemag.

org/sciencenow/2011/06/spiders.html#panel-2.

http://news.sciencemag.org/sciencenow/2011/06/spiders.html#panel-2http://news.sciencemag.org/sciencenow/2011/06/spiders.html#panel-2

2.3. Surface tension: a working definition Chapter 2. Definition and Scaling of Surface Tension

Figure 2.2: a) The free surface between air and water at a molecular scale. b) Surface tension is analogous to a negative surface pressure.

2.3 Surface tension: a working definition

Discussions of the molecular origins of surface or interfacial tension may be found elsewhere (e.g. Israelachvili 1995, Rowlinson & Widom 1982 ). Our discussion follows that of de Gennes, Brochard-Wyart & Quere 2003.

Molecules in a fluid feel a mutual attraction. When this attractive force is overcome by thermal agitation, the molecules pass into a gaseous phase. Let us first consider a free surface, for example that between air and water (Fig. 2.2a). A water molecule in the fluid bulk is surrounded by attractive neighbours, while a molecule at the surface has a reduced number of such neighbours and so in an energetically unfavourable state. The creation of new surface is thus energetically costly, and a fluid system will act to minimize surface areas. It is thus that small fluid bodies tend to evolve into spheres; for example, a thin fluid jet emerging from your kitchen sink will generally pinch off into spherical drops in order to minimize the total surface area (see Lecture 5).

If U is the total cohesive energy per molecule, then a molecule at a free surface will lose U/2 relative to molecules in the bulk. Surface tension is a direct measure of this energy loss per unit area of surface. If the characteristic molecular dimension is R and its area thus R2, then the surface tension is U/(2R)2 . Note that surface tension increases as the intermolecular attraction increases and the molecular size decreases. For most oils, 20 dynes/cm, while for water, 70 dynes/cm. The highest surface tensions are for liquid metals; for example, liquid mercury has 500 dynes/cm. The origins of interfacial tension are analogous. Interfacial tension is a material property of a fluid-fluid interface whose origins lie in the different energy per area that acts to resist the creation of new interface. Fluids between which no interfacial tension arises are said to be miscible. For example, salt molecules will diffuse freely across a boundary between fresh and salt water; consequently, these fluids are miscible, and there is no interfacial tension between them. Our discussion will be confined to immiscible fluid-fluid interfaces (or fluid-gas surfaces), at which an effective interfacial (or surface) tension acts.

Surface tension has the units of force/length or equivalently energy/area, and so may be thought of as a negative surface pressure, or, equivalently, as a line tension acting in all directions parallel to the surface. Pressure is generally an isotropic force per area that acts throughout the bulk of a fluid: small surface element dS will feel a total force p(x)dS owing to the local pressure field p(x). If the surface S is closed, and the pressure uniform, the net pressure force acting on S is zero and the fluid remains static. Pressure gradients correspond to body forces (with units of force per unit volume) within a fluid, and so appear explicitly in the Navier-Stokes equations. Surface tension has the units of force per length, and its action is confined to the free surface. Consider for the sake of simplicity a perfectly flat interface. A surface line element d will feel a total force d owing to the local surface tension (x). If the surface line element is a closed loop C, and the surface tension uniform, the net surface tension force acting on C is zero, and the fluid remains static. If surface tension gradients arise, there may be a net force on the surface element that acts to distort it through driving flow.


2.4. Governing Equations Chapter 2. Definition and Scaling of Surface Tension

2.4 Governing Equations

The motion of a fluid of uniform density and dynamic viscosity is governed by the Navier-Stokes equations, which represent a continuum statement of Newtons laws.

(u

)

+ u u = p+ F + 2 u (2.1) t

u = 0 (2.2)

This represents a system of 4 equations in 4 unknowns (the fluid pressure p and the three components of the velocity field u). Here F represents any body force acting on a fluid; for example, in the presence of a gravitational field, F = g where g is the acceleration due to gravity.

Surface tension acts only at the free surface; consequently, it does not appear in the Navier-Stokes equations, but rather enters through the boundary conditions. The boundary conditions appropriate at a fluid-fluid interface are formally developed in Lecture 3. We here simply state them for the simple case of a free surface (such as air-water, in which one of the fluids is not dynamically significant) in order to get a feeling for the scaling of surface tension. The normal stress balance at a free surface must be balanced by the curvature pressure associated with the surface tension:

n T n = ( n) (2.3) [

1 [

where T = pI + u + (u)T]= pI + 2E is the stress tensor, E = u + (u)T

]is the

2 deviatoric stress tensor, and n is the unit normal to the surface. The tangential stress at a free surface must balance the local surface tension gradient:

n T t = t (2.4)

where t is the unit tangent to the interface.

2.5 The scaling of surface tension

Fundamental Concept The laws of Nature cannot depend on arbitrarily chosen system of units. Any physical system is most succinctly described in terms of dimensionless variables.

Buckinghams Theorem For a system withM physical variables (e.g. density, speed, length, viscosity) describable in terms of N fundamental units (e.g. mass, length, time, temperature), there are M N dimensionless groups that govern the system. E.g. Translation of a rigid sphere through a viscous fluid: Physical variables: sphere speed U and radius a, fluid viscosity and density and sphere drag D; M = 5. Fundamental units: mass M , length L and time T ; N = 3. Buckinghams Theorem: there are M N = 2 dimensionless groups: Cd = D/U2 and Re = Ua/. System is uniquely determined by a single relation between the two: Cd = F (Re). We consider a fluid of density and viscosity = with a free surface characterized by a surface tension . The flow is marked by characteristic length- and velocity- scales of, respectively, a and U , and evolves in the presence of a gravitational field g = gz. We thus have a physical system defined in terms of six physical variables (, , , a, U, g) that may be expressed in terms of three fundamental units: mass, length and time. Buckinghams Theorem thus indicates that the system may be uniquely described in terms of three dimensionless groups. We choose

Ua Inertia Re = = = Reynolds number (2.5)

Viscosity

U2 Inertia Fr = = = Froude number (2.6)

ga Gravity

ga2 Gravity Bo = = = Bond number (2.7)

Curvature


2.5. The scaling of surface tension Chapter 2. Definition and Scaling of Surface Tension

The Reynolds number prescribes the relative magnitudes of inertial and viscous forces in the system, while the Froude number those of inertial and gravity forces. The Bond number indicates the relative importance of forces induced by gravity and surface tension. Note that these two forces are comparable

1/2when Bo = 1, which arises at a lengthscale corresponding to the capillary length: c = (/(g)) . For an air-water surface, for example, 70 dynes/cm, = 1g/cm3 and g = 980 cm/s2, so that c 2mm. Bodies of water in air are dominated by the influence of surface tension provided they are smaller than the capillary length. Roughly speaking, the capillary length prescribes the maximum size of pendant drops that may hang inverted from a ceiling, water-walking insects, and raindrops. Note that as a fluid system becomes progressively smaller, the relative importance of surface tension over gravity increases; it is thus that surface tension effects are critical in many in microscale engineering processes and in the lives of bugs.

Finally, we note that other frequently arising dimensionless groups may be formed from the products of Bo, Re and Fr:

U2a Inertia We = = = Weber number (2.8)

Curvature U Viscous

Ca = = = Capillary number (2.9) Curvature

The Weber number indicates the relative magnitudes of inertial and curvature forces within a fluid, and the capillary number those of viscous and curvature forces. Finally, we note that if the flow is marked by a Marangoni stress of characteristic magnitude /L, then an additional dimensionless group arises that characterizes the relative magnitude of Marangoni and curvature stresses:

a Marangoni Ma = = = Marangoni number (2.10)

L Curvature

We now demonstrate how these dimensionless groups arise naturally from the nondimensionalization of Navier-Stokes equations and the surface boundary conditions. We first introduce a dynamic pressure: pd = p g x, so that gravity appears only in the boundary conditions. We consider the special case of high Reynolds number flow, for which the characteristic dynamic pressure is U2 . We define dimensionless primed variables according to:

a u = Uu , pd = U

2 pd , x = ax , t = t , (2.11) U

where a and U are characteristic lenfth and velocity scales. Nondimensionalizing the Navier-Stokes equations and appropriate boundary conditions yields the following system:

(u

)1

2 + u u = pd + u ,

u = 0 (2.12)

t Re

The normal stress condition assumes the dimensionless form:

1 2 1 E pd + z + n n = n (2.13)

Fr Re We

The relative importance of surface tension to gravity is prescribed by the Bond number Bo, while that of surface tension to viscous stresses by the capillary number Ca. In the high Re limit of interest, the normal force balance requires that the dynamic pressure be balanced by either gravitational or curvature stresses, the relative magnitudes of which are prescribed by the Bond number.

The nondimensionalization scheme will depend on the physical system of interest. Our purpose here was simply to illustrate the manner in which the dimensionless groups arise in the theoretical formulation of the problem. Moreover, we see that those involving surface tension enter exclusively through the boundary conditions.


2.6. A few simple examples Chapter 2. Definition and Scaling of Surface Tension

Figure 2.3: Surface tension may be measured by drawing a thin plate from a liquid bath.

2.6 A few simple examples

Measuring surface tension. Since is a tensile force per unit length, it is possible to infer its value by slowly drawing a thin plate out of a liquid bath and measure the resistive force (Fig. 2.3). The maximum measured force yields the surface tension .

Curvature/ Laplace pressure: consider an oil drop in water (Fig. 2.4a). Work is required to increase the radius from R to R+ dR:

dW = podVo pwdVw + owdA (2.14) ' v ' ' v '

mech. E surface E

where dVo = 4R2dR = dVw and dA = 8RdR.

For mechanical equilibrium, we require dW = (p0 pw)4R2dR + ow8RdR = 0 P = (po pw) = 2ow/R.

Figure 2.4: a) An oil drop in water b) When a soap bubble is penetrated by a cylindrical tube, air is expelled from the bubble by the Laplace pressure.


2.6. A few simple examples Chapter 2. Definition and Scaling of Surface Tension

Note:

1. Pressure inside a drop / bubble is higher than that outside P 2/R smaller bubbles have higher Laplace pressure champagne is louder than beer. Champagne bubbles R 0.1mm, 50 dynes/cm, P 102 atm.

42. For a soap bubble (2 interfaces) P = , so for R 5 cm, 35dynes/cm have P 3105atm. R

More generally, we shall see that there is a pressure jump across any curved interface:

Laplace pressure p = n. Examples:

1. Soap bubble jet - Exit speed (Fig. 2.4b) ( )1/2 ( )

4 470dynes/cm Force balance: p = 4/R 300cm/s airU2 U air R 0.001g/cm33cm

2. Ostwald Ripening: The coarsening of foams (or emulsions) owing to diffusion of gas across interfaces, which is necessarily from small to large bubbles, from high to low Laplace pressure.

23. Falling drops: Force balance Mg airU2a gives vfall speed U ga/air. drop integrity requires airU

2 ga < /a

v

raindrop size a < c = /g 2mm. If a drop is small relative to the capillary length, maintains it against the destabilizing influence of aerodynamic stresses.


3. Wetting Puddles. What sets their size? Knowing nothing of surface chemistry, one anticipates that Laplace pressure balances hydrostatic pressure

J

if /H gH H < c = /g = capillary length.

Note:

1. Drops with R < c remain heavily spherical

2. Large drops spread to depth H c so that Laplace + hydrostatic pressures balance at the drops edge. A volume V will thus spread to a radius R s.t. R2c = V , from which

1/2R = (V/c) .

3. This is the case for H2O on most surfaces, Figure 3.1: Spreading of drops of increasing size. where a contact line exists.

Note: In general, surface chemistry can dominate and one need not have a contact line.

More generally, wetting occurs at fluid-solid contact. Two possibilities exist: partial wetting or total wetting, depending on the surface energies of the 3 interfaces (LV , SV , SL).

Now, just as = LV is a surface energy per area or tensile force per length at a liquid-vapour surface, SL and SV are analogous quantities at solid-liquid and solid-vapour interfaces. The degree of wetting determined by spreading parameters:

S = [Esubstrate]dry [Esubstrate]wet = SV (SL + LV ) (3.1)

where only LV can be easily measured. Total Wetting: S > 0 , e = 0 liquid spreads completely in order to minimize its surface energy. e.g. silicon on glass, water on clean glass.

Note: Silicon oil is more likely to spread than H2O since w 70 dyn/cm > s.o. 20 dyn/cm. Final result: a film of nanoscopic thickness resulting from competition between molecular and capillary forces.

Partial wetting: S < 0, e > 0. In absence of g, forms a spherical cap meeting solid at a contact angle e. A liquid is wetting on a particular solid when e < /2, non-wetting or weakly wetting when e > /2. For H2O, a surface is hydrophilic if e < /2, hydrophobic if e > /2 and superhydrophobic if e > 5/6.

Figure 3.2: The same water drop on hydrophobic and hydrophilic surfaces.

Note: if g = 0, drops always take the form of a spherical cap flattening indicates the effects of gravity.

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4. Youngs Law with Applications Youngs Law: what is the equilibrium contact angle e ? Horizontal force balance at contact line: LV cos e = SV SL

SV SL S cos e = = 1 + (Y oung 1805) (4.1)

LV LV

Note:

1. When S 0, cos e 1 e undefined and spreading results.

2. Vertical force balance not satisfied at contact line dimpling of soft surfaces. E.g. bubbles in paint leave a circular rim.

3. The static contact angle need not take its equilibrium value there is a finite range of possible static contact angles.

Figure 4.1: Three interfaces meet at the contact line. Back to Puddles: Total energy:

1 V 1 E = (SL SV )A+ LV A+ gh

2A = S + gV h (4.2) 2 h 2 v v

surface energy grav. pot. energy

dE 1 1Minimize energy w.r.t. h: = SV h1 2 + gV = 0 when S/h

2 = g dh 2 2 J2S e

J

h0 = g = 2c sin 2 gives puddle depth, where c = /g.

Capillary Adhesion: Two wetted surfaces can stick together with great strength if e < /2, e.g. Fig. 4.2. Laplace Pressure: ( )

1 cos e cos eP = 2R H/2 H

i.e. low P inside film provided e < /2. cos eIf H R, F = R2 2 is the attractive force H

between the plates. Figure 4.2: An oil drop forms a capillary bridge between two glass plates.

E.g. for H2O, with R = 1 cm, H = 5 m and e = 0, one finds P 1/3 atm and an adhesive force F 10N , the weight of 1l of H2O.

Note: Such capillary adhesion is used by beetles in nature.

4.1 Formal Development of Interfacial Flow Problems

Governing Equations: Navier-Stokes. An incompressible, homogeneous fluid of density and viscosity = ( is dynamic and kinematic viscosity) acted upon by an external force per unit volume f evolves according to

u = 0 (continuity) (4.3) (u

)

+ u u = p+ f+ 2 u (Linear momentum conservation) (4.4) t

12

4.1. Formal Development of Interfacial Flow Problems Chapter 4. Youngs Law with Applications

This is a system of 4 equations in 4 unknowns (u1, u2, u3, p). These N-S equations must be solved subject to appropriate BCs. Fluid-Solid BCs: No-slip: u = Usolid. E.g.1 Falling sphere: u = U on sphere surface, where U is the sphere velocity. E.g.2 Convection in a box: u = 0 on the box surface. But we are interested in flows dominated by interfacial effects. Here, in general, one must solve N-S equations in 2 domains, and match solutions together at the interface with appropriate BCs. Difficulty: These interfaces are free to move Free boundary problems.

Figure 4.4: E.g.4 Water waves at an air-water interface.

Figure 4.3: E.g.3 Drop motion within a fluid.

Continuity of Velocity at an interface requires that u = u. And what about p ? Weve seen p /R for a static bubble/drop, but to answer this question in general, we must develop stress conditions at a fluid-fluid interface. Recall: Stress Tensor. The state of stress within an incompressible Newtonian fluid is described by

1 [

the stress tensor: T = pI + 2E where E = (u) + (u)T]is the deviatoric stress tensor. The

2 associated hydrodynamic force per unit volume within the fluid is T . One may thus write N-S eqns in the form: Du = T + f = p+ 2u + f.Dt Now: Tij = force / area acting in the ej direction on a surface with a normal ei.

Note:

1. normal stresses (diagonals) T11, T22, T33 involve both p and ui

2. tangential stresses (off-diagonals) T12, T13, etc., involve only velocity gradients, i.e. viscous stresses

3. Tij is symmetric (Newtonian fluids)

4. t(n) = nT = stress vector acting on a surface with normal n

E.g. Shear flow. Stress in lower boundary is tangential. Force / area on lower boundary:

uTyx = x y |y=0 = k is the force/area that acts on y-surface in x-direction.

Note: the form of T in arbitrary curvilinear coordi- Figure 4.5: Shear flow above a rigid lower bound-

nates is given in the Appendix of Batchelor. ary.


5. Stress Boundary Conditions

Today:

1. Derive stress conditions at a fluid-fluid interface. Requires knowledge of T = pI + 2E

2. Consider several examples of fluid statics

Recall: the curvature of a string under tension may support a normal force. (see right)

5.1 Stress conditions at a fluid-fluid interface

We proceed by deriving the normal and tangential stress boundary conditions appropriate at a fluid-fluid interface characterized by an interfacial tension .

Figure 5.1: String under tension and the influence of gravity.

Figure 5.2: A surface S and bounding contour C on an interface between two fluids. Local unit vectors are n, m and s.

Consider an interfacial surface S bounded by a closed contour C. One may think of there being a force per unit length of magnitude in the s-direction at every point along C that acts to flatten the surface S. Perform a force balance on a volume element V enclosing the interfacial surface S defined by the contour C:

Du

dV = f dV + [t(n) + t(n)] dS + s d (5.1) Dt SV V C

Here indicates arc-length and so d a length increment along the curve C. t(n) = n T is the stress vector, the force/area exerted by the upper (+) fluid on the interface.

[The stress tensor is defined in terms of the local fluid pressure and velocity field as T = pI+ u + (u)T

].

The stress exerted on the interface by the lower (-) fluid is t(n) = n T = n T [

where T = pI + u+ (u)T].

Physical interpretation of terms I

Du dV : inertial force associated with acceleration of fluid in VV Dt

I f dV : body forces acting within V

It(n) dS : hydrodynamic force exerted by upper fluid

S I

t(n) dS : hydrodynamic force exerted by lower fluid S I

s d : surface tension force exerted on perimeter. C

14

5.1. Stress conditions at a fluid-fluid interface Chapter 5. Stress Boundary Conditions

Now if is the characteristic height of our volume V and R its characteristic radius, then the accel-eration and body forces will scale as R2, while the surface forces will scale as R2. Thus, in the limit of 0, the latter must balance.

t(n) + t(n) dS +

s d = 0 (5.2)S C

Now we have thatt(n) = n T , t(n) = n T = n T (5.3)

Moreover, the application of Stokes Theorem (see below) allows us to write

s d =C

S n (S n) dS (5.4)S

where the tangential (surface) gradient operator, defined

S = [I nn] = n (5.5)

n

appears because and n are only defined on the surface S. We proceed by dropping the subscript s on, with this understanding. The surface force balance thus becomes

(

n T n T)

dS = S

n ( n) dS (5.6)S

Now since the surface S is arbitrary, the integrand must vanish identically. One thus obtains the interfacialstress balance equation, which is valid at every point on the interface:Stress Balance Equation

n T n T = n ( n) (5.7)

Interpretation of terms:n T stress (force/area) exerted by + on - (will generally have both and components)

n T stress (force/area) exerted by - on + (will generally have both and components)n ( n) normal curvature force per unit area associated with local curvature of interface, n

tangential stress associated with gradients in

Normal stress balance Taking n(5.7) yields the normal stress balance

n T n n T n = ( n) (5.8)

The jump in the normal stress across the interface is balanced by the curvature pressure.Note: If n = 0, there must be a normal stress jump there, which generally involves both pressure andviscous terms.


6

5.2. Appendix A : Useful identity Chapter 5. Stress Boundary Conditions

Tangential stress balance Taking d(5.7), where d is any linear combination of s and m (any tangentto S), yields the tangential stress balance at the interface:

n T d n T d = d (5.9)

Physical Interpretation

LHS represents the jump in tangential components of the hydrodynamic stress at the interface

RHS represents the tangential stress (Marangoni stress) associated with gradients in , as mayresult from gradients in temperature or chemical composition c at the interface since in general = (, c)

LHS contains only the non-diagonal terms of T - only the velocity gradients, not pressure; thereforeany non-zero at a fluid interface must always drive motion.

5.2 Appendix A : Useful identity

Recall Stokes Theorem:

F d =

n ( F ) dS (5.10)C S

Along the contour C, d = m d, so that we have

F m d =C

n ( F ) dS (5.11)S

Now let F = f b, where b is an arbitrary constant vector. We thus have

(f b) m d =

n ( (f b)) dS (5.12)C S

Now use standard vector identities to see (f b) m = b (f m) and

(f b) = f ( b) b ( f) + b f f b = b ( f) + b f (5.13)

since b is a constant vector. We thus have

b

(f m) d = b

[n ( f) (f) n] dS (5.14)C S

Since b is arbitrary, we thus have

(f m) d =

[n ( f) (f) n] dS (5.15)C S

We now choose f = n, and recall that n m = s. One thus obtains

sd =

[n (n) (n) n] dS = [n n+ n ( n) (n) n] dS.

C S S

We note that n = 0 since must b

e tangent to the surface S and ( 1n) n = 2

(n n) =1

2(1) = 0, and so obtain the desired result:

s d =

[ n ( n)] dS (5.16)C S


5.3. Fluid Statics Chapter 5. Stress Boundary Conditions

5.3 Fluid Statics

We begin by considering static fluid configurations, for which the stress tensor reduces to the form T = pI, so that n T n = p, and the normal stress balance equation (5.8) assumes the simple form:

p p = n (5.17)

The pressure jump across a static interface is balanced by the curvature force at the interface. Now since n T d = 0 for a static system, the tangential stress balance indicates that = 0. This leads to the following important conclusion: There cannot be a static system in the presence of surface tension gradients. While pressure jumps can sustain normal stress jumps across a fluid interface, they do not contribute to the tangential stress jump. Consequently, tangential surface (Marangoni) stresses can only be balanced by viscous stresses associated with fluid motion. We proceed by applying equation (5.17) to describe a number of static situations.

1. Stationary Bubble : We consider a spherical air bubble of radius R submerged in a static fluid. What is the pressure drop across the bubble surface? The divergence in spherical coordinates of F = (Fr, F, F) is given by

1 ( )

1 1 F = r2Fr + (sin F) + F. r2 r r sin r sin

1 2Hence n|S = r2|r=R = so the normal stress jump (5.17) indicates that r2 r R

2 P = p p = (5.18)

R

The pressure within the bubble is higher than that outside by an amount proportional to the surface tension, and inversely proportional to the bubble size. As noted in Lec. 2, it is thus that small bubbles are louder than large ones when they burst at a free surface: champagne is louder than beer. We note that soap bubbles in air have two surfaces that define the inner and outer surfaces of the soap film; consequently, the pressure differential is twice that across a single interface.

2. The static meniscus (e < /2)

Consider a situation where the pressure within a early with z. Such a situation arises in the static static fluid varies owing to the presence of a gravi- meniscus (below). tational field, p = p0 +gz, where p0 is the constant ambient pressure, and g = gz is the grav. acceleration. The normal stress balance thus requires that the interface satisfy the Young-Laplace Equation:

gz = n (5.19)

The vertical gradient in fluid pressure must be balanced by the curvature pressure; as the gradient is constant, the curvature must likewise increase lin- Figure 5.3: Static meniscus near a wall.

The shape of the meniscus is prescribed by two factors: the contact angle between the air-water interface and the wall, and the balance between hydrostatic pressure and curvature pressure. We treat the contact angle e as given; noting that it depends in general on the surface energy. The normal force balance is expressed by the Young-Laplace equation, where now = w air w is the density difference between water and air. We define the free surface by z = (x); equivalently, we define a functional f(x, z) = z (x) that vanishes on the surface. The normal to the surface z = (x) is thus

f z (x)xn = = (5.20)

|f | [1 + (x)2]1/2



As deduced in Appendix B, the curvature of the free surface n, may be expressed as

xx n = xx (5.21)

(1 + 2)3/2x

Assuming that the slope of the meniscus remains sufficiently small, 2x 1, allows one to linearize equation(5.21), so that (5.19) assumes the form

g = xx (5.22)

Applying the boundary condition () = 0 and the contact condition x(0) = cot , and solving (5.22)thus yields

(x) = c cot eex/c (5.23)

where c =

/g is the capillary length. The meniscus formed by an object floating in water is exponen-tial, decaying over a length scale c. Note that this behaviour may be rationalized as follows: the systemarranges itself so that its total energy (grav. potential + surface) is minimized.

3. Floating Bodies

Floating bodies must be supported by some combination of buoyancy and curvature forces. Specifically,since the fluid pressure beneath the interface is related to the atmospheric pressure p0 above the interfaceby

p = p0 + gz + n , (5.24)

one may express the vertical force balance as

Mg = z

pnd = Fb + F .

c

c (5.25)

Cbuo

yancy urvature

The buoyancy force

Fb = z

gzn d = gVb (5.26)C

is thus simply the weight of the fluid displaced above the object and inside the line of tangency (see figurebelow). We note that it may be deduced by integrating the curvature pressure over the contact area Cusing the first of the Frenet-Serret equations (see Appendix C).

F c = z

( n)n d = z C

dt

d = z (t1 t2) = 2 sin (5.27)C d

At the interface, the buoyancy and curvature forces must balance precisely, so the Young-Laplace relationis satisfied:

0 = gz + n (5.28)

Integrating this equation over the meniscus and taking the vertical component yields the vertical forcebalance:

Fmb + Fmc = 0 (5.29)

where

Fmb = z

gzn d = gVm (5.30)Cm

dtFmc = z

( n)n d = z

d = z (t1 tCm Cm d

2) = 2 sin (5.31)

where we have again used the Frenet-Serret equations to evaluate the curvature force.


http:solving(5.22http:5.21),sothat(5.19


Figure 5.4: A floating non-wetting body is supported by a combination of buoyancy and curvature forces, whose relative magnitude is prescribed by the ratio of displaced fluid volumes Vb and Vm.

Equations (5.27-5.31) thus indicate that the curvature force acting on the floating body is expressible in terms of the fluid volume displaced outside the line of tangency:

Fc = gVm (5.32)

The relative magnitude of the buoyancy and curvature forces supporting a floating, non-wetting body is thus prescribed by the relative magnitudes of the volumes of the fluid displaced inside and outside the line of tangency:

Fb Vb = (5.33)

Fc Vm

For 2D bodies, we note that since the meniscus will have a length comparable to the capillary length, 1/2

c = (/(g)) , the relative magnitudes of the buoyancy and curvature forces,

Fb r , (5.34)

Fc c

is prescribed by the relative magnitudes of the body size and capillary length. Very small floating objects (r c) are supported principally by curvature rather than buoyancy forces. This result has been extended to three-dimensional floating objects by Keller 1998, Phys. Fluids, 10, 3009-3010.



Figure 5.5: a) Water strider legs are covered with hair, rendering them effectively non-wetting. The tarsal segment of its legs rest on the free surface. The free surface makes an angle with the horizontal, resulting in an upward curvature force per unit length 2 sin that bears the insects weight. b) The relation between the maximum curvature force Fs = 2P and body weight Fg = Mg for 342 species of water striders. P = 2(L1 + L2 + L3) is the combined length of the tarsal segments. From Hu, Chan & Bush; Nature 424, 2003.

4. Water-walking Insects

Small objects such as paper clips, pins or insects may reside at rest on a free surface provided the curvature force induced by their deflection of the free surface is sufficient to bear their weight (Fig. 5.5a). For example, for a body of contact length L and total mass M , static equilibrium on the free surface requires that:

Mg < 1 , (5.35)

2L sin

where is the angle of tangency of the floating body. This simple criterion is an important geometric constraint on water-walking insects. Fig. 5.5b indicates the dependence of contact length on body weight for over 300 species of water-striders, the most common water walking insect. Note that the solid line corresponds to the requirement (5.35) for static equilibrium. Smaller insects maintain a considerable margin of safety, while the larger striders live close to the edge. The maximum size of water-walking insects is limited by the constraint (5.35).

If body proportions were independent of size L, one would expect the body weight to scale as L3 and

the curvature force as L. Isometry would thus suggest a dependence of the form Fc Fg 1/3

, represented as the dashed line. The fact that the best fit line has a slope considerably larger than 1/3 indicates a variance from isometry: the legs of large water striders are proportionally longer.


5.4. Appendix B : Computing curvatures Chapter 5. Stress Boundary Conditions

5.4 Appendix B : Computing curvatures

We see the appearance of the divergence of the surface normal, n, in the normal stress balance. We proceed by briefly reviewing how to formulate this curvature term in two common geometries.

In cartesian coordinates (x, y, z), we consider a surface defined by z = h(x, y). We define a functional f(x, y, z) = z h(x, y) that necessarily vanishes on the surface. The normal to the surface is defined by

f z hxx hyyn = = (5.36)

|f | ( )1/2 1 + h2 + h2 x y

and the local curvature may thus be computed:

( ) (hxx + hyy) hxxhy

2 + hyyhx2 + 2hxhyhxy

n = (5.37) ( )3/2 1 + h2 + h2 x y

In the simple case of a two-dimensional interface, z = h(x), these results assume the simple forms:

z hxx hxx n = , n = (5.38)

1/2 3/2(1 + h2 ) (1 + h2 )x x

Note that n is dimensionless, while n has the units of 1/L. In 3D polar coordinates (r, , z), we consider a surface defined by z = h(r, ). We define a functional

g(r, , z) = z h(r, ) that vanishes on the surface, and compute the normal:

r 1 hg z hr r n = = , (5.39) )1/2|g| ( 1 h21 + h2 +r r2

from which the local curvature is computed:

2 hrh2 r2hrr hrrh2 + hrhhr n = r r (5.40)

h h2h + hrhtheta rhr

r2 (

1 )1/2

h21 + h2 +r r2

In the case of an axisymmetric interface, z = h(r), these reduce to:

z hrr rhr r2hrr n = , n = (5.41)

1/2 3/2(1 + h2) r2 (1 + h2)r r

5.5 Appendix C : Frenet-Serret Equations

Differential geometry yields relations that are of- Note that the LHS of (5.42) is proportional to the ten useful in computing curvature forces on 2D in- curvature pressure acting on an interface. Therefore terfaces. the net force acting on surface S as a result of cur

vature / Laplace pressures: I I

dtdt F = ( n)n d = d = (t2 t1)( n)n = (5.42) C C d d and so the net force on an interface resulting from dn curvature pressure can be deduced in terms of the

( n) t = (5.43) d geometry of the end points.


6. More on Fluid statics

Last time, we saw that the balance of curvature and hydrostatic pressures requiresg = n = xx

(1+2)3/2.

x

We linearized, assuming x 1, to find (x). Note: we can integrate directly

xxx d 2 d 1

gx = g = 3/2

(1 + 2

dxx)

(

2

)

dx 1/2(1 + 2

x)

1 g2

d 1 1= dx = 1 = 1 sin

2 dx 1/2 1/2x (1 + 2x)

(1 + 2x)

1 sin + g2 =

2(6.1)

Figure 6.1: Calculating the shape and maximal rise height of a static meniscus.

Maximal rise height: At z = h we have = e, so from (6.1)1gh2 = (12

sin e), from which

h =

2c(1 sin e)1/2

where c =

/g (6.2)

Alternative perspective: Consider force balance on the meniscus.Horizontal force balance:

1 sin + gz2 = (6.3)

2horiz. pr

oj

ecti

on of T1 T2

hydrostatic suction

Vertical force balance:

cos =

vert. proj. of T

gzdx (6.4)

1

wei

At x = 0, where = e, gives cos e = weight of fluid d

x

ght

of flu

id

isplaced above z = 0.

Note: cos e = weight of displaced fluid is +/ according to whether e is smaller or larger than .2

Floating Bodies Without considering interfacial effects, one anticipates that heavy things sink and lightthings float. This doesnt hold for objects small relative to the capillary length.Recall: Archimedean force on a submerged body FA =

pndS = gVB .S

In general, the hydrodynamic force acting on a body in a fluidFh =

T ndS, where T = pI+ 2E = pI for static fluid.

S

Here Fh = pndS =S

gzndS = S

g z dV by divergence theorem. This is equal toV

g dV z = gV z = weight of displaced flu

id. The archimedean force can thus support weightV

of a body Mg = BgV if F > B (fluid density larger than body density); otherwise, it sinks.

22

6.1. Capillary forces on floating bodies Chapter 6. More on Fluid statics

Figure 6.2: A heavy body may be supported on a fluid surface by a combination of buoyancy and surface tension.

6.1 Capillary forces on floating bodies

arise owing to interaction of the menisci of floating bodies

attractive or repulsive depending on whether the menisci are of the same or opposite sense

explains the formation of bubble rafts on champagne

explains the mutual attraction of Cheerios and their attraction to the walls

utilized in technology for self-assembly on the microscale

Capillary attraction Want to calculate the attractive force between two floating bodies separated by a distance R. Total energy of the system is given by

f 1 1 h(x)

Etot = dA(R) + dx gzdz (6.5) 0

where the first term in (6.5) corresponds to the total surface energy when the two bodies are a distance R apart, and the second term is the total gravitational potential energy of the fluid. Differentiating (6.5) yields the force acting on each of the bodies:

dEtot(R)F (R) = (6.6)

dR

Such capillary forces are exploited by certain water walking insects to climb menisci. By deforming the free surface, they generate a lateral force that drives them up menisci (Hu & Bush 2005).


7. Spinning, tumbling and rolling drops

7.1 Rotating Drops

We want to find z = h(r) (see right). Normal stress balance on S:

1 2P + 2 r = n

2 "

centrifugal

Nondimensionalize:

( r ) 2

p + 4B0 = n,a ap 2 =

" curvature

3 a centrifugal where p , = = = = 8 curvature Rotational Bond number = const. Define surface functional: f(r, ) = z h(r) vanishes on the surface. Thus

2f zhr (r)r rhr r hrr n = || )3/2 = and n = (1+h2 (r))1/2 r2(1+h2

r r Figure 7.1: The radial profile of a rotating drop.

Brown + Scriven (1980) computed drop shapes and stability for B0 > 0:

1. for < 0.09, only axisymmetric solutions, oblate ellipsoids

2. for 0.09 < < 0.31, both axisymmetric and lobed solutions possible, stable

3. for > 0.31 no stable solution, only lobed forms

Tektites: centimetric metallic ejecta formed from spinning cooling silica droplets generated by meteorite impact.

Q1: Why are they so much bigger than raindrops? V

From raindrop scaling, we expect c but g both , higher by a factor of 10 large tektite size suggests they are not equilibrium forms, but froze into shape during flight.

Q2: Why are their shapes so different from those of raindrops? Owing to high of tektites, the internal dynamics (esp. rotation) dominates the aerodynamics drop shape set by its rotation.

Figure 7.2: The ratio of the maximum radius to the unperturbed radius is indicated as a function of . Stable shapes are denoted by the solid line, their metastable counterparts by dashed lines. Predicted 3-dimensional forms are compared to photographs of natural tektites. From Elkins-Tanton, Ausillous, Bico, Quere and Bush (2003).

24

7.2. Rolling drops Chapter 7. Spinning, tumbling and rolling drops

Light drops: For the case of < 0, < 0, a spinning drop is stabilized on axis by centrifugal pressures.For high ||, it is well described by a cylinder with spherical caps. Drop energy:

1E = I2 + 2rL

2S f

R

a

at

y

ot ional

ur ce energ

K.E.

2

Neglecting the end caps, we write volume V = r2L and moment of inertia I = mr = Lr4.2 2

Figure 7.3: A bubble or a drop suspended in a denser fluid, spinning with angular speed .

The energy per unit drop volume is thus E = 12r2 + 2 .V 4 rMinimizing with respect to r:

31d

( /E) 1 1/3/2= 12r 2 = 0, which occurs when r =

(4 . Now r = V = 42 dr V 2 r 2

)

L

(

2

)

( 3/2

( )

Vonneguts Formula: = 1 2 V3/2

)allows inference of from L, useful technique for small

4 Las it avoids difficulties associated with fluid-solid contact.Note: r grows with and decreases with .

7.2 Rolling drops

Figure 7.4: A liquid drop rolling down an inclined plane.

(Aussillous and Quere 2003 ) Energetics: for steady descent at speed V, MgV sin =Rate of viscousdissipation= 2

(u)2dV , where Vd is the dissipation zone, so this sets V = V/R is the angularVd

speed. Stability characteristics different: bioconcave oblate ellipsoids now stable.


8. Capillary Rise Capillary rise is one of the most well-known and vivid illustrations of capillarity. It is exploited in a number of biological processes, including drinking strategies of insects, birds and bats and plays an important role in a number of geophysical settings, including flow in porous media such as soil or sand.

Historical Notes:

Leonardo da Vinci (1452 - 1519) recorded the effect in his notes and proposed that mountain streams may result from capillary rise through a fine network of cracks

Jacques Rohault (1620-1675): erroneously suggested that capillary rise is due to suppresion of air circulation in narrow tube and creation of a vacuum

Geovanni Borelli (1608-1675): demonstrated experimentally that h 1/r

Geminiano Montanari (1633-87): attributed circulation in plants to capillary rise

Francis Hauksbee (1700s): conducted an extensive series of capillary rise experiments reported by Newton in his Opticks but was left unattributed

James Jurin (1684-1750): an English physiologist who independently confirmed h 1/r; hence Jurins Law.

Consider capillary rise in a cylindrical tube of inner radius a (Fig. 8.2)

Recall: Spreading parameter: S = SV (SL + LV ).

We now define Imbibition / Impregnation parameter: I = SV SL = LV cos via force balance at contact line. Note: in capillary rise, I is the relevant parameter, since motion of the contact line doesnt change the energy of the liquid-vapour interface.

Imbibition Condition: I > 0.

Note: since I = S + LV , the imbibition condition I > 0 is always more easily met than the spreading condition, S > 0 Figure 8.1: Capillary rise and fall in a tube for two most liquids soak sponges and other porous me- values of the imbibition parameter I: dia, while complete spreading is far less common. I > 0 (left) and I < 0 (right).

26

Chapter 8. Capillary Rise

We want to predict the dependence of rise height H on both tube radius a and wetting properties. We do so by minimizing the total system energy, specifically the surface and gravitational potential energies. The energy of the water column:

E = (SL SV ) 2aH + 1 ga2H2 = 2aHI +

1 ga2H2

; 2 2surface energy

;

grav.P.E.

will be a minimum with respect to H when dE = 0dH

2SV SL 2 I H = = , from which we deduce ga ga

LV cos Jurins Law H = 2 (8.1)

gr

Note:

1. describes both capillary rise and descent: sign of H depends on whether > /2 or < /2

2. H increases as decreases. Hmax for = 0

3. weve implicitly assumed R H & R lC .

The same result may be deduced via pressure or force arguments. By Pressure Argument Provided a c, the meniscus will take the form

aof a spherical cap with radius R = . Therefore cos

cos cos pA = pB 2 = p0 2 = p0 gH a a 2 cos

H = as previously. ga By Force Argument The weight of the column supported by the tensile force acting along the contact line: a2Hg = 2a (SV SL) = 2a cos , from which Jurins Law again follows.

Figure 8.2: Deriving the height of capillary rise in a tube via pressure arguments.


8.1. Dynamics Chapter 8. Capillary Rise

8.1 Dynamics

The column rises due to capillary forces, its rise being resisted by a combination of gravity, viscosity, fluid I

inertia and dynamic pressure. Conservation of momentum dictates d (m(t)z(t)) = FTOT + vv ndA,dt S 2where the second term on the right-hand side is the total momentum flux, which evaluates to a2z = mz,

so the force balance on the column may be expressed as

1

2 m + ma z = 2a cos mg a2 z 2az v (8.2) 2; ; ; ; ;

Inertia capillary force Added mass weight ;

viscous force dynamic pressure

where m = a2z. Now assume the flow in the tube is fully developed Poiseuille flow, which will be 2

( )aestablished after a diffusion time = . Thus, u(r) = 2z 1 a

r 22 , and F = a

2z is the flux along the

tube. du

4The stress along the outer wall: = z.|r=a = dr a Finally, we need to estimate ma, which will dominate the dynamics at short time. We thus estimate the (

1change in kinetic energy as the column rises from z to z+z. Ek = mU2), where m = mc+m0+m2

(mass in the column, in the spherical cap, and all the other mass, respectively). In the column,mc = a2z,

2 u = U . In the spherical cap, m0 = a3, u = U . In the outer region, radial inflow extends to , but

3 u(r) decays. Volume conservation requires: a2U = 2a2ur(a) ur(a) = U/2.

a aContinuity thus gives: 2a2ur(a) = 2r2ur(r) ur(r) = r2

2 ur(a) = 2r

2

2 U . eff U2 1

I

(r)2Thus, the K.E. in the far field: 1 m = ur dm, where dm = 2r2dr.

2 2 a

Hence 1 ( )21 a2

m = eff U 2r2dr = U2 2r2 a

1 1 1

= a4 dr = a3 2r2 2a

Now

1)U2

1 (mc + m0 + m + m2UU =Ek =

2 2 1 1 ( eff )U2 = mc + mc + m0 + m 2UU =

2 2 1 ( )

U2 (

2 1 )

= a2z + a2z+ a3+ a3 UU2 3 2 Figure 8.3: The dynamics of capillary rise.

7 4Substituting for m = a2z, ma = a3 (added mass) and v = z into (8.2) we arrive at 6 a

( )7 2 cos 1 8zz

z + a z = z2 gz (8.3) 6 a 2 a2

The static balance clearly yields the rise height, i.e. Jurins Law. But how do we get there?


8.1. Dynamics Chapter 8. Capillary Rise

Inertial Regime

1. the timescale of establishment of Poiseuille flow is = 4a

2 , the time required for bound

ary effects to diffuse across the tube

2. until this time, viscous effects are negligible and the capillary rise is resisted primarily by Figure 8.4: The various scaling regimes of capillary fluid inertia rise.

7 2 cos Initial Regime: z 0, z 0, so the force balance assumes the form 6 a az = We thus infer

6 cos z(t) = t2 .7 a2

7 (

7 )

2 cos Once z a, one must also consider the column mass, and so solve z + a z = . As the col6 6 a

2 2 2 cos umn accelerates from z = 0, z becomes important, and the force balance becomes: 1 z = 2 a

( )1/2 4 cos z = U = is independent of g, .a

( )1/2 4 cos z = t.a

Viscous Regime (t ) Here, inertial effects become negligible, so the force balance assumes the form: 2 cos

8zz 8zz 2 cos ga2 (

H )

gz = 0. We thus infer H z = , where H = , z = 1a a2 ga2 ga 8 z 8H Nondimensionalizing: z = z/H, t = t/ , = ;ga2

1 z 1 t We thus have z = z dt

=

dz = (1 1z )dz

= z ln(1 z ). 1 1z

Note: at t , z 1.

Early Viscous Regime: When z 1, we consider ln(z 1) = z 1 z 2and so infer z = 2t .

2 [ ]1/2 a cos Redimensionalizing thus yields Washburns Law : z =

2 t

Note that z is independent of g.

Late Viscous Regime: As z approaches H, z 1. Thus, we consider t = [z ln(1z )] = ln(1z ) and so infer z = 1 exp(t ).

2 cos 8H Redimensionalizing yields z = H [1 exp(t/)], where H = and = ga ga2 . 2

( )1/2 4a 4 cos 4a 2

Note: if rise timescale = , inertia dominates, i.e. H Uintertial = inertial a overshoot arises, giving rise to oscillations of the water column about its equilibrium height H.

cos Wicking In the viscous regime, we have 2 = a 8zza2 + g. What if the viscous stresses dominate gravity? This may arise, for example, for predominantly horizontal flow (Fig. 8.5).

2a cos 1 d 2Force balance: = zz = z z = 8 2 dt

( )1/2 a cos

2 t t (Washburns Law).

Note: Front slows down, not due to g, but owing to increasing viscous dissipation with increasing col- Figure 8.5: Horizontal flow in a small tube. umn length.


9. Marangoni Flows Marangoni flows are those driven by surface gradients. In general, surface tension depends on both the temperature and chemical composition at the interface; consequently, Marangoni flows may be generated by gradients in either temperature or chemical composition at an interface. We previously derived the tangential stress balance at a free surface:

n T t = t , (9.1)

where n is the unit outward normal to the surface, and t is any unit tangent vector. The tangential component of the hydrodynamic stress at the surface must balance the tangential stress associated with gradients in . Such Marangoni stresses may result from gradients in temperature or chemical composition at the interface. For a static system, since n T t = 0, the tangential stress balance equation indicates that: 0 = . This leads us to the following important conclusion:

There cannot be a static system in the presence of surface tension gradients. While pressure jumps can arise in static systems characterized by a normal stress jump across a fluid interface, they do not contribute to the tangential stress jump. Consequently, tangential surface stresses can only be balanced by viscous stresses associated with fluid motion. Thermocapillary flows: Marangoni flows induced by temperature gradients (T ).

d Note that in general < 0 Why? A warmer gas phase has more liquid molecules, so the creation of dT surface is less energetically unfavourable; therefore, is lower. Approach Through the interfacial BCs (and (T )s appearance therein), N-S equations must be coupled

Figure 9.1: Surface tension of a gas-liquid interface decreases with temperature since a warmer gas phase contains more suspended liquid molecules. The energetic penalty of a liquid molecule moving to the interface is thus decreased.

to the heat equation T

+ u T = 2T (9.2) t

30

Chapter 9. Marangoni Flows

Note:

1. the heat equation must be solved subject to appropriate BCs at the free surface. Doing so can be complicated, especially if the fluid is evaporating.

Ua 2. Analysis may be simplified when the Peclet number Pe = 1. Nondimensionalize (9.2): a

x = ax , t = t , u = Uu to get U ( )

PeT

+ u T = 2T (9.3) t

Note: Ua

Pe = Re Pr = 1 if Re 1, so one has 2T = 0. The Prandtl number Pr = O(1) for many common (e.g. aqeous) fluids. E.g.1 Thermocapillary flow in a slot (Fig.9.2a)

d T USurface Tangential BCs = = viscous stress U 1 H.L dT L H L

Figure 9.2: a) Thermocapillary flow in a slot b) Thermal convection in a plane layer c) Thermocapillary drop motion.

E.g.2 Thermocapillary Drop Motion (Young, Goldstein & Block 1962) can trap bubbles in gravitational field via thermocapillary forces. (Fig.9.2c). E.g.3 Thermal Marangoni Convection in a Plane Layer (Fig.9.2b). Consider a horizontal fluid layer heated from below. Such a layer may be subject to either buoyancy- or Marangoni-induced convection. Recall: Thermal buoyancy-driven convection (Rayleigh-Bernard) (T ) = 0 (1 + (T T0)), where is the thermal expansivity. Consider a buoyant blob of characteristic scale d. Near the onset of convection,

gTd2 one expects it to rise with a Stokes velocity U g d2 = . The blob will rise, and so convection

d d will occur, provided its rise time rise = = is less than the time required for it to lose its heat U gTd2 d2 and buoyancy by diffusion, diff = .

dif f gTd3 Criterion for Instability: Ra > Rac 103, where Ra is the Rayleigh number. rise

Note: for Ra < Rac, heat is transported solely through diffusion, so the layer remains static. For Ra > Rac, convection arises. The subsequent behaviour depends on Ra and Pr. Generally, as Ra increases, steady convection rolls time-dependency chaos turbulence.



Thermal Marangoni Convection

Arises because of the dependence of on temperature: (T ) = 0 (T T0) Mechanism:

Imagine a warm spot on surface prompts surface divergence upwelling.

Upwelling blob is warm, which reinforces the perturbation provided it rises before losing its heat via diffusion.

U Balance Marangoni and viscous stress: d d

d d Rise time: U

Diffusion time diff = d 2

dif f Td Criterion for instability: Ma > Mac, where Ma is the Marangoni number. rise

Note:

1. Since Ma d and Ra d3, thin layers are most unstable to Marangoni convection.

2. Benards original experiments performed in millimetric layers of spermaceti were visualizing Marangoni convection, but were misinterpreted by Rayleigh as being due to buoyancy not recognized until Block (Nature 1956).

3. Pearson (1958) performed stability analysis with flat surface deduced Mac = 80 .

4. Scriven & Sternling (1964) considered a deformable interface, which renders the system unstable at all Ma. Downwelling beneath peaks in Marangoni convection, upwelling between peaks in RayleighBenard convection (Fig. 9.3a).

5. Smith (1966) showed that the destabilizing influence of the surface may be mitigated by gravity. d dT 2Stability Criterion: < gd thin layers prone to instability. dT dz 3



E.g.4 Marangoni Shear Layer (Fig. 9.3) Lateral leads to Marangoni stress shear flow. The resulting T (x, y) may destabilize the layer to

Figure 9.3: a) Marangoni convection in a shear layer may lead to transverse surface waves or streamwise rolls (c). Surface deflection may accompany both instabilities (b,d).

Marangoni convection. Smith & Davis (1983ab) considered the case of flat free surface. System behaviour depends on Pr = /. Low Pr: Hydrothermal waves propagate in direction of . High Pr: Streamwise vortices (Fig. 9.3c). Hosoi & Bush (2001) considered a deformable free surface (Fig. 9.3d)

E.g.5 Evaporatively-driven convection e.g. for an alcohol-H2O solution, evaporation affects both the alcohol concentration c and temperature . The density (c, ) and surface tension (c, ) are such that < 0, < 0, d < 0, d < 0. Evaporation c d dc results in surface cooling and so may generate either Rayleigh-Benard or Marangoni thermal convection. Since it also induces a change in surface chemistry, it may likewise generate either Ra B or Marangoni chemical convection.

E.g.6 Coffee Drop Marangoni flows are responsible for the ring-like stain left by a coffee drop.

coffee grounds stick to the surface

evaporation leads to surface cooling, which is most pronounced near the edge, where surface area per volume ratio is highest

resulting thermal Marangoni stresses drive radial outflow on surface radial ring

Figure 9.4: Evaporation of water from a coffee drop drives a Marangoni flow.


10. Marangoni Flows II

10.1 Tears of Wine

The first Marangoni flow considered was the tears of wine phenomenon (Thomson 1885 ), which actually predates Marangonis first published work on the subject by a decade. The tears of wine phenomenon is readily observed in a wine glass following the establishment of a thin layer of wine on the walls of the glass.

An illustration of the tears of wine phenomenon is shown in Fig. 10.1. Evaporation of alcohol occurs everywhere along the free surface. The alcohol concentration in the thin layer is thus reduced relative to that in the bulk owing to the enhanced surface area to volume ratio. As surface tension decreases with alcohol concentration, the surface tension is higher in the thin film than the bulk; the associated Marangoni stress drives upflow throughout the thin film. The wine climbs until reaching the top of the film, where it accumulates in a band of fluid that thickens until eventually becoming gravitationally unstable and releasing the tears of wine. The tears or legs roll back to replenish the bulk reservoir, but with fluid that is depleted in alcohol.

The flow relies on the transfer of chemical potential energy to kinetic and ultimately gravitational potential energy. The process continues until the fuel for the process, the alcohol is completely depleted. For certain liquors (e.g. port), the climbing film, a Marangoni shear layer, goes unstable to streamwise vortices and an associated radial corrugation - the tear ducts of wine (Hosoi & Bush, JFM 2001). When the descending tears reach the bath, they appear to recoil in response to the abrupt change in . The tears or legs of wine are taken by sommeliers to be an indicator of the quality of wine.

Figure 10.1: The tears of wine. Fluid is drawn from the bulk up the thin film adjoining the walls of the glass by Marangoni stresses induced by evaporation of alcohol from the free surface.

34

10.2. Surfactants Chapter 10. Marangoni Flows II

Figure 10.2: a) A typical molecular structure of surfactants. b) The typical dependence of on surfactant concentration c.

10.2 Surfactants

Surfactants are molecules that have an affinity for interfaces; common examples include soap and oil. Owing to their molecular structure (e.g. a hydrophilic head and hydrophobic tail, Fig. 10.2a), they find it energetically favourable to reside at the free surface. Their presence reduces the surface tension; consequently, gradients in surfactant concentration result in surface tension gradients. Surfactants thus generate a special class of Marangoni flows. There are many different types of surfactants, some of which are insoluble (and so remain on the surface), others of which are soluble in the suspending fluid and so diffuse into the bulk. For a wide range of common surfactants, surface tension is a monotonically decreasing function of until a critical micell concentration (CMC) is achieved, beyond CMC there is no further dependence of on (Fig. 10.2b). Surfactant properties:

Diffusivity prescribes the rate of diffusion, Ds (bulk diffusivity Db), of a surfactant along an interface

Solubility prescribes the ease with which surfactant passes from the surface to the bulk. An insoluble surfactant cannot dissolve into the bulk, must remain on the surface.

Volatility prescribes the ease with which surfactant sublimates.

Theoretical Approach: because of the dependence of on the surfactant concentration, and the appearance of in the boundary conditions, N-S equations must be augmented by an equation governing the evolution of . In the bulk,

c = Db

2+ u c c (10.1) t

The concentration of surfactant on a free surface evolves according to

2+ s (us) + (s n) (u n) = J (, Cs) + Ds (10.2) st

where us is the surface velocity, s is the surface gradient operator and Ds is the surface diffusivity of the surfactant (Stone 1999). J is a surfactant source term associated with adsorption onto or desorption from the surface, and depends on both the surface surfactant concentration and the concentration in the bulk Cs. Tracing the evolution of a contaminated free surface requires the use of Navier-Stokes equations, relevant boundary conditions and the surfactant evolution equation (10.2). The dependence of surface tension on surfactant concentration, (), requires the coupling of the flow field and surfactant field. In certain special cases, the system may be made more tractable. For example, for insoluble surfactants, J = 0. Many surfactants have sufficiently small Ds that surface diffusivity may be safely neglected.



Figure 10.3: The footprint of a whale, caused by the whales sweeping biomaterial to the sea surface. The biomaterial acts as a surfactant in locally suppressing the capillary waves evident elsewhere on the sea surface. Observed in the wake of a whale on a Whale Watch from Boston Harbour.

Special case: expansion of a spherical surfactant-laden bubble. dR d dR = 2 dR + ( n) ur = 0. Here n = 2/R, ur = so + 2 = 0 d 4R2 =const., so t dt dt R dt R the surfactant is conserved.

Marangoni Elasticity The principal dynamical influence of surfactants is to impart an effective elasticity to the interface. One can think of a clean interface as a slippery trampoline that resists deformation through generation of normal curvature pressures. However, such a surface cannot generate traction on the interface. However, a surface-laden interface, like a trampoline, resists surface deformation as does a clean interface, but can also support tangential stresses via Marangoni elasticity. Specifically, the presence of surfactants will serve not only to alter the normal stress balance (through the reduction of ), but also the tangential stress balance through the generation of Marangoni stresses.

The presence of surfactants will act to suppress any fluid motion characterized by non-zero surface divergence. For example, consider a fluid motion characterized by a radially divergent surface motion. The presence of surfactants results in the redistribution of surfactants: is reduced near the point of divergence. The resulting Marangoni stresses act to suppress the surface motion, resisting it through an effective surface elasticity. Similarly, if the flow is characterized by a radial convergence, the resulting accumulation of surfactant in the region of convergence will result in Marangoni stresses that serve to resist it. It is this effective elasticity that gives soap films their longevity: the divergent motions that would cause a pure liquid film to rupture are suppressed by the surfactant layer on the soap film surface.

The ability of surfactant to suppress flows with non-zero surface divergence is evident throughout the natural world. It was first remarked upon by Pliny the Elder, who rationalized that the absence of capillary waves in the wake of ships is due to the ships stirring up surfactant. This phenomenon was also widely known to spear-fishermen, who poured oil on the water to increase their ability to see their prey, and by sailors, who would do similarly in an attempt to calm troubled seas. Finally, the suppression of capillary waves by surfactant is at least partially responsible for the footprints of whales (see Fig. 10.3). In the wake of whales, even in turbulent roiling seas, one seas patches on the sea surface (of characteristic width 5-10m) that are perfectly flat. These are generally acknowledged to result from the whales sweeping biomaterial to the surface with their tails, this biomaterial serving as a surfactant that suppresses capillary waves.



Surfactants and a murder mystery. From Nature, 22, 1880 : In the autumn of 1878 a man committed a terrible crime in Somerset, which was for some time involved

in deep mystery. His wife, a handsome and decent mulatto woman, disappeared suddenly and entirely from

sight, after going home from church on Sunday, October 20. Suspicion immediately fell upon the husband,

a clever young fellow of about thirty, but no trace of the missing woman was left behind, and there seemed

a strong probability that the crime would remain undetected. On Sunday, however, October 27, a week

after the woman had disappeared, some Somerville boatmen looking out towards the sea, as is their custom,

were struck by observing in the Long Bay Channel, the surface of which was ruffled by a slight breeze, a

streak of calm such as, to use their own illustration, a cask of oil usually diffuses around it

when in the water. The feverish anxiety about the missing woman suggested some strange connection

between this singular calm and the mode of her disappearance. Two or three days after - why not sooner

I cannot tell you - her brother and three other men went out to the spot where it was observed, and from

which it had not disappeared since Sunday, and with a series of fish-hooks ranged along a long line dragged

the bottom of the channel, but at first without success. Shifting the position of the boat, they dragged a

little further to windward, and presently the line was caught. With water glasses the men discovered that

it had caught in a skeleton which was held down by some heavy weight. They pulled on the line; something

suddenly gave was, and up came the skeleton of the trunk, pelvis, and legs of a human body, from which

almost every vestige of flesh had disappeared, but which, from the minute fragments remaining, and the

terrible stench, had evidently not lain long in the water. The husband was a fisherman, and Long Bay

Channel was a favourite fishing ground, and he calculated, truly enough, that the fish would very soon

destroy all means of identification; but it never entered into his head that as they did so their ravages,

combined with the process of decomposition, set free the matter which was to write the traces of

his crime on the surface of the water. The case seems to be an exceedingly interesting one; the calm

is not mentioned in any book on medical jurisprudence that I have, and the doctors seem not to have had

experience of such an occurrence. A diver went down and found a stone with a rope attached, by which

the body had been held down, and also portions of the scalp and of the skin of the sole of the foot, and of

clothing, by means of which the body was identified. The husband was found guilty and executed.


10.3. Surfactant-induced Marangoni flows Chapter 10. Marangoni Flows II

Figure 10.4: The soap boat. A floating body (length 2.5cm) contains a small volume of soap, which serves as its fuel in propelling it across the free surface. The soap exits the rear of the boat, decreasing the local surface tension. The resulting fore-to-aft surface tension gradient propels the boat forward. The water surface is covered with Thymol blue, which parts owing to the presence of soap, which is thus visible as a white streak.

10.3 Surfactant-induced Marangoni flows

1. Marangoni propulsion

Consider a floating body with perimeter C in contact with the free surface, which we assume for the sake of simplicity to be flat. Recalling that may be though of as a force per unit length in a direction tangent to the surface, we see that the total surface tension force acting on the body is:

Fc = sd (10.3) C

where s is the unit vector tangent to the free surface and normal to C, and d is an increment of arc length along C. If is everywhere constant, then this line integral vanishes identically by the divergence theorem. However, if = (x), then it may result in a net propulsive force. The soap boat may be simply generated by coating one end of a toothpick with soap, which acts to reduce surface tension (see right). The concomitant gradient in surface tension results in a net propulsive force that drives the boat away from the soap. We note that an analogous Marangoni propulsion technique arises in the natural world: certain water-walking insects eject surfactant and use the resulting surface tension gradients as an emergency mechanism for avoiding predation. Moreover, when a pine needle falls into a lake or pond, it is propelled across the surface in an analogous fashion owing to the influence of the resin at its base decreasing the local surface tension.

2. Soap film stability

Pinching a film increases the surface area, decreases and so increases . Fluid is thus drawn in and the film is stabilized by the Marangoni elasticity.

Figure 10.5: Fluid is drawn to a pinched area of a soap film through induced Marangoni stresses.


10.4. Bubble motion Chapter 10. Marangoni Flows II

3. Vertical Soap Film

Vertical force balance: gh(z) = d . The weight of a soap dz film is supported by Marangoni stress.

Internal dynamics: note that film is dynamic (as are all Marangoni flows), if it were static, its max height would be c. It is constantly drying due to the influence of gravity.

d du On the surface: balance of Marangoni and visdz dx cous stresses.

d2 u Inside: g dx2 Gravity-viscous.

Figure 10.6: The weight of a vertical soap film is supported by Marangoni 10.4 Bubble motion stresses on its surface.

Theoretical predictions for the rise speed of small drops or bubbles do not adequately describe observations. Specifically, air bubbles rise at low Reynolds number at rates appropriate for rigid spheres with equivalent buoyancy in all but the most carefully cleaned fluids. This discrepancy may be rationalized through consideration of the influence of surfactants on the surface dynamics. The flow generated by a clean spherical bubble or radius a rising at low Re = Ua/ is intuitively obvious. The interior flow is toroidal, while the surface motion is characterized by divergence and convergence at, respectively, the leading and trailing surfaces. The presence of surface contamination changes the flow qualitatively.

The effective surface elasticity imparted by the surfactants acts to suppress the surface motion. Surfactant is generaly swept to the trailing edge of the bubble, where it accumulates, giving rise to a local decrease in surface tension. The resulting for-to-aft surface tension gradient results in a Marangoni stress that resists surface motion, and so rigidifies the bubble surface. The air bubble thus moves as if its surface were stagnant, and it is thus that its rise speed is commensurate with that predicted for a rigid sphere: the no-slip boundary condition is more appropriate than the free-slip. Finally, we note that the characteristic Marangoni stress /a is most pronounced for small bubbles. It is thus that the influence of surfactants is greatest on small bubbles.

Figure 10.7: A rising drop or bubble (left) is marked by internal circulation in a clean system that is absent in a contaminated, surfactant-laden fluid (right). Surfactant sticks to the surface, and the induced Marangoni stress rigidifies the drop surface.


11. Fluid Jets

11.1 The shape of a falling fluid jet

Consider a circular orifice of a radius a ejecting a flux Q of fluid density and kinematic viscosity (see Fig. 11.1). The resulting jet accelerates under the influence of gravity gz. We assume that the jet Reynolds number Re = Q/(a) is sufficiently high that the influence of viscosity is negligible; furthermore, we assume that the jet speed is independent of radius, and so adequately described by U(z). We proceed by deducing the shape r(z) and speed U(z) of the evolving jet.

Applying Bernoullis Theorem at points A and B:

1 U2 + gz + PA =

1 U2(z) + PB (11.1)

2 0 2

The local curvature of slender threads may be expressed in terms of the two principal radii of curvature, R1 and R2:

1 1 1 n = + (11.2)

R1 R2 r

Thus, the fluid pressures within the jet at points A and B may be simply related to that of the ambient, P0:

, PB P0 + (11.3) PA P0 +

a r

Substituting into (11.1) thus yields

1 1 U2 + gz + P0 + = U

2(z) + P0 + (11.4) 2 0 a 2 r

from which one finds 1/2

U(z) 2 z 2 ( a) = 1 + + 1 (11.5)

U0 Fr a We r acc. due to g dec. due to Figure 11.1: A fluid jet extruded

where we define the dimensionless groups from an orifice of radius a accelerates under the influence of U2 INERTIA

0Fr = = = Froude Number (11.6) gravity. Its shape is influenced

ga GRAV ITY both by the gravitational acceler

U0

2a INERTIA ation g and the surface tension . CURV ATURE

We = = = Weber Number (11.7) Note that gives rise to a gradi-

Now flux conservation requires that ent in curvature pressure within r the jet, /r(z), that opposes the

Q = 2 U(z)r(z)dr = a2U0 = r2U(z) (11.8) acceleration due to g.

0

from which one obtains ( )

1/2 [( )

]

1/4 r(z) U0 2 z 2 a

= = 1 + + 1 (11.9) a U(z) Fr a We r

This may be solved algebraically to yield the thread shape r(z)/a, then this result substituted into (11.5) to deduce the velocity profile U(z). In the limit of We , one obtains

( )1/4 ( )1/2

r 2gz U(z) 2gz = 1 + , = 1 + (11.10)

a U0

2 U0 U02

40

11.2. The Plateau-Rayleigh Instability Chapter 11. Fluid Jets

11.2 The Plateau-Rayleigh Instability

We here summarize the work of Plateau and Rayleigh on the instability of cylindrical fluid jets bound by surface tension. It is precisely this Rayleigh-Plateau instability that is responsible for the pinch-off of thin water jets emerging from kitchen taps (see Fig. 11.2).

The equilibrium base state consists of an infinitely long quiescent cylindrical inviscid fluid column of radius R0, density and surface tensi

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